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Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)

Distributed cavity phase calculation for a rectangular Ramsey cavity in NRC-FCs1

L. Marmet, P. Dubé, Nicolas A. Shtin, R. J. Mauricio Lopez

Published Jul 1, 2013• 1 citation
Publisher

Abstract

A preliminary calculation of the frequency shifts resulting from the distributed phase of a rectangular cavity is used to model frequency measurements of the fountain clock NRC-FCs1. The calculation is based on a numerical evaluation of the cavity field made with a finite element EM solver. The frequency shift is obtained by solving the Bloch-equations numerically along the path of the atoms on the free-fall trajectories through the cavity. We study the frequency shifts as a function of the launch direction, the tilt of the physics package and offsets in the position of the MOT relative to the axis defined by the state-selection cavity and the Ramsey cavity.

DOI
10.1109/eftf-ifc.2013.6702294

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A preliminary calculation of the frequency shifts resulting from the distributed phase of a rectangular cavity is used to model frequency measurements of the fountain clock NRC-FCs1. The calculation is based on a numerical evaluation of the cavity field made with a finite element EM solver. The frequency shift is obtained by solving the Bloch-equations numerically along the path of the atoms on the free-fall trajectories through the cavity. We study the frequency shifts as a function of the launch direction, the tilt of the physics package and offsets in the position of the MOT relative to the axis defined by the state-selection cavity and the Ramsey cavity.
We discuss the power dependence of distributed cavity phase errors for cylindrical TE011cavities in laser-cooled atomic fountain clocks. The azimuthally symmetric phase variations produce a surprisingly large distributed cavity phase error for two 2pi, 4pi, and 6pi pulses. This is due to the correlation between the transverse variation of the Rabi frequency over the cavity aperture and a quadratic density variation of the atomic sample, along with the symmetry of the longitudinal phase variation in the cavity. We show that the large azimuthally symmetric fields and phase shifts near the walls of the endcap holes produce very small errors at optimal power for a uniform wall resistance. We also show the power variation for higher order azimuthal variations m=1, 2, and 4. These may be caused by fountain tilts, nonuniform detection of atoms, and asymmetries in the laser trapping and cooling of the atoms. We demonstrate that distributed cavity phase errors in physical cavities may have no variation with the microwave power. A combination of rigorous calculations of cavity losses, measurements of power dependence, the atomic distributions, and fountain tilts, and electrical measurements that show the lower limit of the cavity Q and the cavity symmetry, should provide stringent limits on distributed cavity phase errors for current atomic clocks
We demonstrate a convenient experimental method, that facilitates faster and more accurate evaluation of the distributed cavity phase, a leading systematic effect in atomic fountain clocks. We expand on earlier work by Nemitz et al., which showed that displacement of the atoms from the microwave cavity axis can lead to an asymmetry in resonances observed for hyperfine transitions with ΔmF= 1. By applying an additional small transverse magnetic field, with controlled magnitude and direction, we are able to extract from the measured asymmetry the atom cloud's crossing position to within a fraction of a millimeter.
We give a review of recent advances in the accuracy of atomic fountain clocks. The new features are improved experimental and theoretical treatments of distributed cavity phase shifts and a full modeling of microwave lensing. Here we highlight that state selection is often highly non-uniform when Rabi flopping is used instead of an interrupted adiabatic frequency chirp. We show that detuning the state-selection microwave field as well as changing its amplitude can give a much more homogeneous state selection. A spatially inhomogeneous state selection corrupts the density extrapolation because the density distribution changes, which also mixes effects from distributed cavity phase and microwave lensing. This is expected to be important for the microgravity clock PHARAO on ACES, and can also improve fountain clocks.
We investigate theoretically and by numerical simulations the effects of the atomic recoil on the frequency and fringe contrast of microwave atomic frequency standards. Such effects arise because of the influence of external degrees of freedom on the phases and the spatial positions of the interfering wave packets. We show under which conditions such effects lead to a frequency shift and examine the loss of contrast of the interference signal (Ramsey fringes). We use an interferometric description to model the frequency standards, treating simultaneously internal and external degrees of freedom. We consider multiple momentum exchanges in 3D between the atoms and the standing microwave field inside the cavity leading to a large number of wave packets that leave the cavity with different momenta. At detection, these wave packets interfere giving rise to Ramsey fringes that are modulated by the interference of the external states of the wave packets. Similar treatments have proved efficient when treating ID recoil effects in standing laser waves. We apply this theoretical model numerically to laser cooled atomic fountain frequency standards and investigate the behaviour of the observed signal as a function of the frequency standard parameters (microwave power, launching height, etc.). Finally, we discuss possible experimental methods that could be used to observe such effects.
We report a realization of Zacharias's 1953 proposal for observing a Ramsey resonance in an atomic fountain. Launched upward from a moving optical molasses where they have been cooled to ∼ 5 μK, cesium atoms pass once through a microwave cavity, continue to the summit of their trajectory, then fall again through the same cavity, completing the separated-fields interaction. The atoms spend 0.25 s in free flight above the cavity. Linewidth (2 Hz) and S/N imply a stability of 3·10 -12 τ -1/2 , at least as good as in existing Cs clocks, with eventual expected improvements of 10 2 .